hedwards: I had a hard time pulling up anything authoritative, but I did find a reference to Nelson buying press by spending a lot of money engaging in unseemly activities, but it wasn't at all clear whether it was a turn of phrase that would have been used at the time or by the person writing the article in the present day.
cjrgreen: This is also described as "buying press coverage", but that's also an older (and literal) expression.
Newsweek wrote (2010) that Obama "bought his own press" when he thought he had enough influence to push for drastic healthcare reforms. It's more widely used in Hollywood, where it's been applied to acts of directorial hubris perpetrated by everyone from George Lucas to Gregg Araki.
I was a bit surprised that there wasn't anything that I could find that predated that. It's sometimes interesting to look up phrases that you see all over the place. But, to find that there wasn't anything available online was somewhat odd, usually you can find something.
I would wager it's because the original phrase was pretty generic and this version was only recently coined.